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Cohomological equation and cocycle rigidity of discrete parabolic actions

Approved for Public Release; Distribution Unlimited. Case Number 18-1082. The first author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the authors. ©2018 The MITRE Corporation. All rights reserved.
1 Based on research supported by NSF grant DMS-1700837

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  • We study the cohomological equation for discrete horocycle maps on $ {\rm SL}(2, \mathbb{R}) $ and $ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $ {\rm SL}(2, \mathbb{R}) $. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for $ {\rm SL}(2, \mathbb{R}) $ in which all cases of irreducible, unitary representations of $ {\rm SL}(2, \mathbb{R}) $ can be studied simultaneously.

    Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

    Mathematics Subject Classification: 37A17, 37A20.


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